21 PRODUCT OF MEASURES 233 To each real-valued function /on R we associate the function /0 on R2 defined by MX, y) =/(*) if y < xtf0(xt y) = /0>) if y *> x. Show that/is v-integrable if and only if /o is (A (x) /z)-integrable, and that in this case \fdv = JT/0 d\ dp,. (Prove the result first for characteristic functions of intervals.) Hence deduce the formula jf(x) dw(x) = J/(xXje-) du(x) + J/(*Mjc+) dv(x). In particular, if u and v are continuous on R, we have the formula of integration by parts: Ŗn(x) dv(x) = u(b)v(b) - u(a)v(a) - JV*) du(x). Consider the case where u and v are constant on each interval [Ģ, n + 1 [ for each integer Ģ ^ 0 ("Abel's partial summation formula"). 7. Let;? be a finite real number j>l, let X and Y be locally compact spaces, A a positive measure on X and p a positive measure on Y. Let /^ 0 be a function on X x Y such that/ and /p are (A (x) /j,)-integrable. Prove that (For each xeX, apply Holder's inequality (13.11, Problem 12(a)) to the function y*~+f(x, y), in the form/(x, y) = g(x, y)(\ fp(x, y) d\(x)jP\ where q is the exponent conjugate to p.) 8. Let X/ (1 <:i<^n) be locally compact spaces and ^ a positive measure on X, for 1 <zi<:n. For each /, let E, denote the product f"] X, . Let// be a function >0 which n n is measurable with respect to u = Ž a* on X = 0 X( , and does not depend on xt . If iĢi 1*1 fl~l is integrable with respect to the measure Ž pj, for 1 <] A: ^w, show that the function O/' *s /^-integrable and that '' /i/2 '' 'fn dpi dfj,2 - - dp,n < I lp[ J where Jfc = I /jj"1 dpv - d(j,k-t djjLk+ i t//>tn for I ^k^n. (By induction on Ģ, using the Lebesgue-Fubini theorem and Holder's inequality.) Deduce that if A is a /x-measurable subset of X, and At its projection on Et, and if At is integrable and of measure mi with respect to the measure (x) /z, on Et, then A is /x-integrable and /x(A) ^ (mi mn)1/(l'1"1>. Generalize to the case in which, instead of considering the products of the Xi n 1 at a time, we consider the (J) products of the X< p at a time, and integrate over X a product of (J) functions ^0, each of which depends on only p of the variablestion belonging to JT(Y) is the uniform limit of functions belonging